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Cram
In statistics, Cramér's V (sometimes referred to as Cramér's phi or Cramers C and denoted as φ''c) is a popular measure of association between two nominal variables, giving a value between 0 and +1 (inclusive). It is based on Pearson's chi-squared statistic and was published by Harald Cramér in 1946.Cramér, Harald. 1946. ''Mathematical Methods of Statistics. Princeton: Princeton University Press, p282. ISBN 0-691-08004-6 Usage and interpretation φ''c'' is the intercorrelation of two discrete variablesSheskin, David J. (1997). Handbook of Parametric and Nonparametric Statistical Procedures. Boca Raton, Fl: CRC Press. and may be used with variables having two or more levels. φ''c'' is a symmetrical measure, it does not matter which variable we place in the columns and which in the rows. Also, the order of rows/columns doesn't matter, so φ''c'' may be used with nominal data types or higher (ordered, numerical, etc) Cramér's V may also be applied to goodness of fit chi-squared models when there is a 1×k table (e.g: r''=1). In this case ''k is taken as the number of optional outcomes and it functions as a measure of tendency towards a single outcome. Cramér's V varies from 0 (corresponding to no association between the variables) to 1 (complete association) and can reach 1 only when the two variables are equal to each other. φ''c''2 is the mean square canonical correlation between the variables . In the case of a 2×2 contingency table Cramér's V is equal to the Phi coefficient. Note that as chi-squared values tend to increase with the number of cells, the greater the difference between r'' (rows) and ''c (columns), the more likely φc will tend to 1 without strong evidence of a meaningful correlation. Calculation Cramér's V is computed by taking the square root of the chi-squared statistic divided by the sample size and the length of the minimum dimension (k'' is the smaller of the number of rows ''r or columns c''). The formula for the φ''c coefficient is: : \phi_c = \sqrt{\frac{\varphi^2}{(k-1)}} = \sqrt{ \frac{\chi^2}{N(k - 1)}} where: * \varphi^2 is the phi coefficient. * \chi^2 is derived from Pearson's chi-squared test * N is the grand total of observations and * k being the number of rows or the number of columns, whichever is less. The p-value for the significance of φ''c'' is the same one that is calculated using the Pearson's chi-squared test . The formula for the variance of φ''c'' is known.Liebetrau, Albert M. (1983). Measures of association. Newbury Park, CA: Sage Publications. Quantitative Applications in the Social Sciences Series No. 32. (pages 15–16) Unlike the contingency tablethe value of V is relatively independent of the number of columns or rows. See also Other measures of correlation for nominal data: * The phi coefficient * Tschuprow's T * The uncertainty coefficient * The Lambda coefficient Other related articles: * Effect size References * Cramér, H. (1999). Mathematical Methods of Statistics, Princeton University Press External links * A Measure of Association for Nonparametric Statistics (Alan C. Acock and Gordon R. Stavig Page 1381 of 1381–1386) * Nominal Association: Phi, Contingency Coefficient, Tschuprow's T, Cramer's V, Lambda, Uncertainty Coefficient Category:Categorical data Category:Statistical dependence Category:Statistical ratios Category:Summary statistics for contingency tables